For complex parameters λ and s, consider the Lerch transcendent Φ(λ, s, z) = ∑∞ k=0 λ (k + z)−s as a function of the complex variable z. We analyze the asymptotic behavior of this function as Re s→ −∞.

The sums of three trigonometric series with logarithmic coefficients are derived by extending an approach first utilized Lerch. By applying Frullani's theorem to two these series, non-trivial integrals involving hyperbolic functions evaluated in terms the digamma function.

Keywords: Trigonometric functions Hurwitz zeta function Legendre chi function Lerch zeta function Bernoulli polynomials Euler polynomials a b s t r a c t In this sequel to our recent note [D. Cvijović, Values of the derivatives of the cotangent at rational multiples of π, Appl. Math. Lett. it is shown, in a unified manner, by making use of some basic properties of certain special functions, suc...

We construct bivariate polynomial approximations of the Lerch function, which for certain specialisations of the variables and parameters turn out to be HermitePadé approximants either of the polylogarithms or of Hurwitz zeta functions. In the former case, we recover known results, while in the latter the results are new and generalise some recent works of Beukers and Prévost. Finally, we make ...